Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Thursday, March 30, 2006

Uniform Derandomization Assumptions

In 1986 during the prehistory of Hardness vs. Randomness, Sipser
showed
that if time does not have nontrivial space simulations one can
derandomize. Given a (later-proved) assumption about extractor constructions

If there is a constant c such that DTIME(2cn) is not
contained in DSPACE(2.99cn) then P=RP.

Later results used hardness against nonuniform circuits to derandomize
complexity classes. For example Impagliazzo and Wigderson show

If E does not have circuits of size
2o(n) then P=BPP.

where E=DTIME(2O(n)).

I recently discovered that Peter Bro Miltersen in his derandomization
survey (page 56) notes that you can use a uniform assumption, a
weaker version of Sipser's assumption.

If E is not contained is DSPACE(2o(n))
then E does not have circuits of size
2o(n) and thus P=BPP.

Miltersen's proof works by searching all circuits, using the local
checkability of E to verify the correctness of the circuit.

You can even assume the circuits have access to QBF or
Σ2p gates, the later an assumption we
needed in a recent
paper.
Saying E is not contained in subexponential space is so much
nicer than saying E does not have nonuniform subexponential-size
circuits with Σ2p gates.

Technical Note: For all of the above you need to assume the
separations at all input lengths.